Mix 1/2 gallon water with 3/4 cup salt, stirring for salt to dissolve. Soak cucumbers in salt water about 30 minutes. (Be careful not to soak them too long.) Remove cucumbers and rinse. Cut about 1/4 inch from each end of cucumbers. Cut cucumbers in half in middle, not lengthwise. Hold cucumber facing circular middle. Cut in half lengthwise, leaving about 1/2 inch at end uncut. Cut in half lengthwise again, perpendicular to your first cut, leaving the end uncut. You should have 4 semi-equal parts of cucumber, cut but still attached. Repeat with remaining cucumbers.

Set 3 (1-quart) jars on work surface. Using your fingers, separate cucumber quarters and stuff mix into cucumbers. Divide evenly among jars, pressing cucumbers down firmly into jars. Stir 1 teaspoon sugar into 1/3 cup water until sugar is completely dissolved. Pour sugar water over cucumbers.